Kay Rülling: Ramification theory of reciprocity sheaves

Oberseminar, December 16, 2021

Abstract.

Reciprocity sheaves were introduced by Kahn-Saito-Yamazaki as a generalization of homotopy invariant sheaves a la Voevodsky. They are in particular Nisnevich sheaves on all smooth schemes over a perfect base field $k$, which have the additional property, that it is possible to measure the degree of ramification at infinity of a given section on a smooth open. The interest comes from the fact that this notion of ramification is defined in a motivic way. Examples of reciprocity sheaves are: smooth commutative k-group schemes, the isomorphism classes of fppf-torsors under finite commutative $k$-group schemes, Kähler differentials, de Rham-Witt differentials, and many more.

In this talk I will explain the ramification filtration of reciprocity sheaves, their link to classical defined ramification filtrations and recent structure results  obtained in joint work with Shuji Saito. The main results are:  Zariski-Nagata purity and a Abbes-(Takeshi)Saito formula. In the course of the proof we construct for any reciprocity sheaf a pairing which generalizes and refines the pairing from Kato-Saito’s higher dimensional geometric class field theory. This work relies heavily on the work of Binda-Rülling-Saito on the cohomology of reciprocity sheaves.